220 research outputs found
Constructing Sobol' sequences with better two-dimensional projections
Direction numbers for generating Sobol' sequences that satisfy the so-called Property A in up to 1111 dimensions have previously been given in Joe and Kuo [ACM Trans. Math. Software, 29 (2003), pp. 49–57]. However, these Sobol' sequences may have poor two-dimensional projections. Here we provide a new set of direction numbers alleviating this problem. These are obtained by treating Sobol' sequences in d dimensions as (t, d)-sequences and then optimizing the t-values of the two-dimensional projections. Our target dimension is 21201
Component-by-component construction of good intermediate-rank lattice rules
It is known that the generating vector of a rank-1 lattice rule can be constructed component-by-component to achieve strong tractability error bounds in both weighted Korobov spaces and weighted Sobolev spaces. Since the weights for these spaces are nonincreasing, the first few variables are in a sense more important than the rest. We thus propose to copy the points of a rank-1 lattice rule a number of times in the first few dimensions to yield an intermediate-rank lattice rule. We show that the generating vector (and in weighted Sobolev spaces, the shift also) of an intermediate-rank lattice rule can also be constructed component-by-component to achieve strong tractability error bounds. In certain circumstances, these bounds are better than the corresponding bounds for rank-1 lattice rules
Lattice rules with random achieve nearly the optimal error independently of the dimension
We analyze a new random algorithm for numerical integration of -variate
functions over from a weighted Sobolev space with dominating mixed
smoothness and product weights
, where the functions are continuous and
periodic when . The algorithm is based on rank- lattice rules
with a random number of points~. For the case , we prove that
the algorithm achieves almost the optimal order of convergence of
, where the implied constant is independent of
the dimension~ if the weights satisfy . The same rate of convergence holds for the more
general case by adding a random shift to the lattice rule with
random . This shows, in particular, that the exponent of strong tractability
in the randomized setting equals , if the weights decay fast
enough. We obtain a lower bound to indicate that our results are essentially
optimal. This paper is a significant advancement over previous related works
with respect to the potential for implementation and the independence of error
bounds on the problem dimension. Other known algorithms which achieve the
optimal error bounds, such as those based on Frolov's method, are very
difficult to implement especially in high dimensions. Here we adapt a
lesser-known randomization technique introduced by Bakhvalov in 1961. This
algorithm is based on rank- lattice rules which are very easy to implement
given the integer generating vectors. A simple probabilistic approach can be
used to obtain suitable generating vectors.Comment: 17 page
Multi-level higher order QMC Galerkin discretization for affine parametric operator equations
We develop a convergence analysis of a multi-level algorithm combining higher
order quasi-Monte Carlo (QMC) quadratures with general Petrov-Galerkin
discretizations of countably affine parametric operator equations of elliptic
and parabolic type, extending both the multi-level first order analysis in
[\emph{F.Y.~Kuo, Ch.~Schwab, and I.H.~Sloan, Multi-level quasi-Monte Carlo
finite element methods for a class of elliptic partial differential equations
with random coefficient} (in review)] and the single level higher order
analysis in [\emph{J.~Dick, F.Y.~Kuo, Q.T.~Le~Gia, D.~Nuyens, and Ch.~Schwab,
Higher order QMC Galerkin discretization for parametric operator equations} (in
review)]. We cover, in particular, both definite as well as indefinite,
strongly elliptic systems of partial differential equations (PDEs) in
non-smooth domains, and discuss in detail the impact of higher order
derivatives of {\KL} eigenfunctions in the parametrization of random PDE inputs
on the convergence results. Based on our \emph{a-priori} error bounds, concrete
choices of algorithm parameters are proposed in order to achieve a prescribed
accuracy under minimal computational work. Problem classes and sufficient
conditions on data are identified where multi-level higher order QMC
Petrov-Galerkin algorithms outperform the corresponding single level versions
of these algorithms. Numerical experiments confirm the theoretical results
On the expected uniform error of geometric Brownian motion approximated by the L\'evy-Ciesielski construction
It is known that the Brownian bridge or L\'evy-Ciesielski construction of
Brownian paths almost surely converges uniformly to the true Brownian path. In
the present article the focus is on the error. In particular, we show for
geometric Brownian motion that at level , at which there are points
evaluated on the Brownian path, the expected uniform error has an upper bound
of order , or equivalently, . This upper bound matches the known order for the expected uniform error
of the standard Brownian motion. We apply the result to an option pricing
example
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